Problem Solving

sxjain3 wrote:Cafe is closer to Dan's house by 2 miles. so if distance between cafe and karen's house is x, then distance between cafe and Dan's house is x-2.


(Karen)-------(X)-------(Cafe)
\ |
\ |
\ |
(10) (x-2)
\ |
\ |
\ |
(Dan)

Its a Right angle triangle, so using Pyto therom.

x^2 + (x-2)^2 = 10^2

the equation can be then solved to get the values of 2 sides as 6 and 8. cafe is closer the Dan's house than Karen's, hence distace between cafe and laren's house is 8.

well played.

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IMPORTANT BACKGROUND INFORMATION

this is the point at which i come on stage, waving a flag and yelling the following incantation:

memorize the famous pythagorean triples.

these are the ones i call 'celebrity triangles', because they look nice and pop up all over the place:
3-4-5 and 5-12-13 are the 'a-list celebrity triangles'
8-15-17 and 7-24-25 are the 'b-list celebrity triangles'

the 'a-list' triangles, just like actual a-list celebrities, pop up in different forms so they're harder to recognize.** so in other words, not only should you look for 3-4-5 triangles, but also 6-8-10 triangles, 9-12-15 triangles, etc. also, in addition to 5-12-13, you should keep an eye out for 10-24-26 triangles.
in addition, the a-list triangles have impostors;** if you see a right triangle with a 3 and a 5 on it, for example, make sure that the 3 is a leg and the 5 is the hypotenuse. if the 3 and 5 are legs, then the hypotenuse is √34, not 4.

the 'b-list' triangles (again just like the corresponding celebrities), because they're more obscure, can walk around in public because not very many people recognize them. in other words, if you see an 8-15-17 triangle, then the side lengths will probably be actually 8, actually 15, and actually 17. same goes for the 7-24-25 (and the c-list triangles like 9-40-41 and 20-21-29, if you're really hardcore).

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if you're familiar with the backgroud information above, then you can solve this problem in less than half a minute (depending on drawing speed), using the following reasoning:
* draw it
* hey look, i need a hypotenuse of 10 and legs that differ by 2
* let me search my mental inventory of celebrity triangles
* 6-8-10!
* done.

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**this is probably the best analogy i've ever concocted in my life.